We show how to do semiclassical nonperturbative computations within the worldline approach to quantum field theory using "worldline instantons". These worldline instantons are classical solutions to the Euclidean worldline loop equations of motion, and are closed spacetime loops parametrized by the proper-time. Specifically, we compute the imaginary part of the one loop effective action in scalar QED using "worldline instantons", for a wide class of inhomogeneous electric field backgrounds. We treat both time dependent and space dependent electric fields, and note that temporal inhomogeneities tend to shrink the instanton loops, while spatial inhomogeneities tend to expand them. This corresponds to temporal inhomogeneities tending to enhance local pair production, with spatial inhomogeneities tending to suppress local pair production. We also show how the worldline instanton technique extends to spinor QED.

We study electron-positron pair creation from the Dirac vacuum induced by a strong and slowly varying electric field (Schwinger effect) which is superimposed by a weak and rapidly changing electromagnetic field (dynamical pair creation). In the sub-critical regime where both mechanisms separately are strongly suppressed, their combined impact yields a pair creation rate which is dramatically enhanced. Intuitively speaking, the strong electric field lowers the threshold for dynamical particle creation -or, alternatively, the fast electromagnetic field generates additional seeds for the Schwinger mechanism. These findings could be relevant for planned ultra-high intensity lasers.PACS numbers: 12.20. Ds, 11.15.Tk, 11.27.+d. As first realized by Dirac [1], a consistent relativistic quantum description of electrons necessarily involves negative energy levels, which -in the Dirac-sea pictureare filled up in the vacuum state. This entails the striking possibility of pulling an electron out of the vacuum by means of some external influence, such as a (classical) electromagnetic field [2], where the remaining hole in the Dirac sea is then associated with a positron. Of course, to create such an electron-positron pair out of the vacuum, one has to overcome the energy gap of 2mc 2 between the filled and the empty levels. There are basically two main mechanisms for doing so: In a strong electric field E over a sufficiently long distance L, "virtual" electron-positron pair fluctuations may gain this energy when qEL ≥ 2mc2 . This pair creation process is called the Schwinger mechanism [3,4] and can be understood as tunneling through the classically forbidden region (energy gap). Thus it is suppressed exponentially O(exp{−πE S /E}) for weak fields E, where E S = m 2 c 3 /( q) is the Schwinger critical field. For E ≃ E S , the work done by separating the electron-positron pair over a Compton wavelength is of the order of the energy gap 2mc2 . Alternatively, a classical time-dependent electromagnetic field will also create electron-positron pairs in general (dynamical pair creation). However, if the frequency ω of the external field is not large enough, ω < 2mc 2 , these non-adiabatic corrections correspond to higher-order (i.e., multi-photon) processes and are also suppressed exponentially exp{− O(1/ω)} for small ω [5]. These pair-production processes are fundamental predictions of quantum electrodynamics (QED), but only the multi-photon production process has so far been observed experimentally: the positron data taken at the SLAC E-144 experiment have convincingly been explained by nphoton production with n ≃ 5 [6]. However, a verification of the Schwinger mechanism has still remained an experimental challenge [7]. Since the Schwinger mechanism is non-perturbative in the field, its discovery would help exploring the non-perturbative realm of quantum field theory in a controlled fashion. Here, we propose a new mechanism which can help to overcome the strong exponential suppression. The basic idea is similar in spirit to ide...

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This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigmamodels, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Écalle resummation. The ideas from QFT use continuity on R 1 ×S 1 L , i.e, the absence of any phase transition as N → ∞ or rapid-crossovers for finite-N , and the small-L weak coupling limit to render the semiclassical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. There exists a resurgent behavior in the semi-classical trans-series analysis of the QFT, whereby subleading orders of exponential terms mix in a systematic way, canceling all ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Θ angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

I present a pedagogical review of Heisenberg-Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.Dedicated to the memory of Ian Kogan, a great physicist and friend, whose enthusiasm for life and science is sorely missed.

In a previous paper [1], it was shown that the worldline expression for the nonperturbative imaginary part of the QED effective action can be approximated by the contribution of a special closed classical path in Euclidean spacetime, known as a worldline instanton. Here we extend this formalism to compute also the prefactor arising from quantum fluctuations about this classical closed path. We present a direct numerical approach for determining this prefactor, and we find a simple explicit formula for the prefactor in the cases where the inhomogeneous electric field is a function of just one spacetime coordinate. We find excellent agreement between our semiclassical approximation, conventional WKB, and recent numerical results using numerical worldline loops.

In these lectures I review classical aspects of the self-dual Chern-Simons systems which describe charged scalar fields in 2 + 1 dimensions coupled to a gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These self-dual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol'nyi bound which is saturated by solutions to a set of first-order self-duality equations. In the nonrelativistic case the self-dual potential is quartic, the system possesses a dynamical conformal symmetry, and the self-dual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic self-duality equations are integrable and all finite charge solutions may be found. In the relativistic case the self-dual potential is sixth order and the self-dual Lagrangian may be embedded in a model with an extended supersymmetry. The self-dual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the self-dual nature of the potential.

Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.